Optimal. Leaf size=8 \[ \frac {1}{2 \cos ^{-1}(x)^2} \]
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Rubi [A] time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4642} \[ \frac {1}{2 \cos ^{-1}(x)^2} \]
Antiderivative was successfully verified.
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Rule 4642
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1-x^2} \cos ^{-1}(x)^3} \, dx &=\frac {1}{2 \cos ^{-1}(x)^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 8, normalized size = 1.00 \[ \frac {1}{2 \cos ^{-1}(x)^2} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1-x^2} \cos ^{-1}(x)^3} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.90, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.93, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 7, normalized size = 0.88
method | result | size |
derivativedivides | \(\frac {1}{2 \arccos \relax (x )^{2}}\) | \(7\) |
default | \(\frac {1}{2 \arccos \relax (x )^{2}}\) | \(7\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.99, size = 6, normalized size = 0.75 \[ \frac {1}{2 \, \arccos \relax (x)^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.35, size = 6, normalized size = 0.75 \[ \frac {1}{2\,{\mathrm {acos}\relax (x)}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.68, size = 7, normalized size = 0.88 \[ \frac {1}{2 \operatorname {acos}^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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